I’m in the process of building my freeplay cm3 and I’ve managed to damage not one but two LCD screens. On the first one, I accidentally slightly tore the ribbon cable near the 40 pin header and on the second one, the adhesive tape tore, causing the leads to no longer connect.

On your first photo, it looks like pin 40 is affected. Are there multiple pins that are affected or just the last one (40)?

On our Freeplay CM3 PCB, pin 39 and 40 are both connected to 3.3V. 38 and 37 are connected to GND.

On the LCD itself, pins 39, 40, 37, and 37 are all separate. IF pin 40 was the only one that was damaged, and IF you could do some fine-pitch soldering, you could, in theory, bridge 39 and 40 together a bit further down the ribbon. You'd have to scratch off the solder mask first. This would be a pretty advanced repair.

Another advanced repair could be to scratch off the solder mask from the finger of 40 and the trace of 40 and solder a jumper wire in between the two.

As for the second screen, I don't think I've seen that before. How did that happen?

I have seen a Gameboy repair where you take a hot soldering iron and iron on top of the LCD ribbon cable where it attaches. I guess that would look something like this. https://youtu.be/-Cj5Pzuz06M?t=513 I have NO IDEA if that would work on this LCD that you're having the problem with.